The domain of definition of the function $\mathrm{f}(x)$ given by the equation $2^x+2^y=2$ is

The domain of definition of the function $\mathrm{f}(x)$ given by the equation $2^x+2^y=2$ is
  1. $0 \lt x \leqslant 1$
  2. $0 \leqslant x \leqslant 1$
  3. $-\infty \lt x \leqslant 0$
  4. $-\infty \lt x \lt 1$

Solution

$\begin{aligned} & 2^x+2^y=2 \\ & \Rightarrow 2^y=2-2^x \\ & \Rightarrow y=\log _2\left(2-2^x\right) \text { is defined, if } 2-2^x\gt0 \\ & \Rightarrow 2^x \lt 2 \\ & \Rightarrow 2^{x-1} \lt 1 \\ & \Rightarrow x-1 \lt 0 \\ & \Rightarrow-\infty \lt x \lt 1\end{aligned}$

Asked in: MHT CET 2024 (09 May Shift 2)

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